Prior Probability Density Function Research Articles

Probabilistic Prediction of Floating Offshore Wind Turbine Platform Motions via Uncertainty Quantification and Information Integration

The accurate prediction of short-term platform motions in a real environment is crucial for the safe design, operation, and maintenance of floating offshore wind turbines (FOWTs). Numerical simulations of motions are typically associated with high uncertainties due to abstracted theoretical models, empirical parameters, initial environment parameters, etc. Therefore, it is necessary to integrate other sources of information associated with less uncertainty, e.g., monitoring data, for accurate predictions. In this paper, we propose a probabilistic prediction based on the Bayesian approach that logically integrates motion monitoring data with simulated motion predictions of FOWTs, considering uncertainties in the environment model, structural properties, motion prediction method, monitoring data, etc. The approach consists of constructing a prior probability density function (PDF) of a random variable (which characterizes the largest value of the initial motion response) via numerical simulations and a likelihood function based on platform motion monitoring data and deriving a posterior PDF of the random variable by Bayesian updating. Then, posterior distributions of short-term extreme motion responses are derived using the posterior PDF of the random variable, representing lower uncertainty and improved accuracy. A Metropolis–Hastings algorithm is adopted to obtain PDFs of complex probability distributions. The effectiveness of the approach is demonstrated on a real FOWT platform in Scotland. The proposed probabilistic prediction approach results in posterior distributions of short-term extreme platform motions associated with less uncertainty and higher accuracy, which is attributed to integrating prior knowledge with monitoring data.

Efficient reliability updating methods based on Bayesian inference and sequential learning Kriging

Reliability updating is an effective tool for reappraising reliability level of system when new observation information is obtained. The adaptive Kriging based reliability updating method (RUAK) inserts the adaptive Kriging into traditional simulation method to improve the efficiency of reliability updating. However, an identical candidate sampling pool is used to simultaneously estimate the prior failure probability and the posterior one in RUAK, which leads to a waste of computational resources in case of significant difference between the importance regions in estimation of prior and posterior failure probabilities. To overcome this disadvantage, an efficient reliability updating framework based on Bayesian inference and sequential learning Kriging is proposed in this paper. In the proposed method, two candidate sampling pools respectively for estimating the prior and posterior failure probabilities are separately constructed by prior probability density function (PDF) and posterior PDF obtained by Bayesian inference. Then, the Kriging model is established and sequentially refined in these two candidate sampling pools to accurately estimate the corresponding failure probabilities. Through combining different simulation methods with the proposed framework, the Monte Carlo simulation based and importance sampling based sequential learning Kriging methods are respectively developed for reliability updating.

Generative models and Bayesian inversion using Laplace approximation

The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows expert knowledge or physical constraints to be formulated in a probabilistic fashion and plays an important role for the success of the inference. Recently, Bayesian inverse problems were solved using generative models as highly informative priors. Generative models are a popular tool in machine learning to generate data whose properties closely resemble those of a given database. Typically, the generated distribution of data is embedded in a low-dimensional manifold. For the inverse problem, a generative model is trained on a database that reflects the properties of the sought solution, such as typical structures of the tissue in the human brain in magnetic resonance imaging. The inference is carried out in the low-dimensional manifold determined by the generative model that strongly reduces the dimensionality of the inverse problem. However, this procedure produces a posterior that does not admit a Lebesgue density in the actual variables and the accuracy attained can strongly depend on the quality of the generative model. For linear Gaussian models, we explore an alternative Bayesian inference based on probabilistic generative models; this inference is carried out in the original high-dimensional space. A Laplace approximation is employed to analytically derive the prior probability density function required, which is induced by the generative model. Properties of the resulting inference are investigated. Specifically, we show that derived Bayes estimates are consistent, in contrast to the approach in which the low-dimensional manifold of the generative model is employed. The MNIST data set is used to design numerical experiments that confirm our theoretical findings. It is shown that the approach proposed can be advantageous when the information contained in the data is high and a simple heuristic is considered for the detection of this case. Finally, the pros and cons of both approaches are discussed.

Fatigue Reliability Based on Predicted Posterior Stress Ranges Determined from Strain Measurements of Wind Turbine Support Structures

In the present paper, an approach for updating the continuous stress range distribution of a welded connection of a wind turbine support structure with predicted information from strain measurements is presented. Environmental conditions, such as wind or, in offshore fields, waves and currents, in combination with rotor excitations generate cyclic stresses affecting the reliability of welded joints of the support structure over the service life. Using strain measurements, these conditions can be monitored, and the resulting stress ranges, under consideration of measurement, mechanical and material uncertainties, can be reconstructed. These stress ranges can be used as an input for updating the prior probability density function (PDF) of the stress ranges predicted by the overall dynamics and a detailed design analysis. Applying Bayesian probability theory and decision theoretical implications, the predicted posterior probability density of the stress ranges is calculated based on the design information and uncertainties. This approach is exemplified, and it is shown how the predicted stress ranges and the design stress ranges are distributed. The prior and the predicted posterior stress ranges are used for a reliability calculation for potentially entering a pre-posterior decision analysis.

A Joint-Parameter Estimation and Bayesian Reconstruction Approach to Low-Dose CT.

Most penalized maximum likelihood methods for tomographic image reconstruction based on Bayes' law include a freely adjustable hyperparameter to balance the data fidelity term and the prior/penalty term for a specific noise-resolution tradeoff. The hyperparameter is determined empirically via a trial-and-error fashion in many applications, which then selects the optimal result from multiple iterative reconstructions. These penalized methods are not only time-consuming by their iterative nature, but also require manual adjustment. This study aims to investigate a theory-based strategy for Bayesian image reconstruction without a freely adjustable hyperparameter, to substantially save time and computational resources. The Bayesian image reconstruction problem is formulated by two probability density functions (PDFs), one for the data fidelity term and the other for the prior term. When formulating these PDFs, we introduce two parameters. While these two parameters ensure the PDFs completely describe the data and prior terms, they cannot be determined by the acquired data; thus, they are called complete but unobservable parameters. Estimating these two parameters becomes possible under the conditional expectation and maximization for the image reconstruction, given the acquired data and the PDFs. This leads to an iterative algorithm, which jointly estimates the two parameters and computes the to-be reconstructed image by maximizing a posteriori probability, denoted as joint-parameter-Bayes. In addition to the theoretical formulation, comprehensive simulation experiments are performed to analyze the stopping criterion of the iterative joint-parameter-Bayes method. Finally, given the data, an optimal reconstruction is obtained without any freely adjustable hyperparameter by satisfying the PDF condition for both the data likelihood and the prior probability, and by satisfying the stopping criterion. Moreover, the stability of joint-parameter-Bayes is investigated through factors such as initialization, the PDF specification, and renormalization in an iterative manner. Both phantom simulation and clinical patient data results show that joint-parameter-Bayes can provide comparable reconstructed image quality compared to the conventional methods, but with much less reconstruction time. To see the response of the algorithm to different types of noise, three common noise models are introduced to the simulation data, including white Gaussian noise to post-log sinogram data, Poisson-like signal-dependent noise to post-log sinogram data and Poisson noise to the pre-log transmission data. The experimental outcomes of the white Gaussian noise reveal that the two parameters estimated by the joint-parameter-Bayes method agree well with simulations. It is observed that the parameter introduced to satisfy the prior's PDF is more sensitive to stopping the iteration process for all three noise models. A stability investigation showed that the initial image by filtered back projection is very robust. Clinical patient data demonstrated the effectiveness of the proposed joint-parameter-Bayes and stopping criterion.

Meta-analysis of the experimental coefficient of variation in wheat using the Bayesian and Frequentist approaches

A large set of variables is assessed for progeny selection in a plant-breeding program and other agronomic fields. The meta-analysis of the coefficient of variation (CVe) produces information for researchers and breeders on the experimental quality of trials. This analysis can also be applied in the decision-making process of the experimental plan regarding the experimental design, the number of repetitions, and the treatments and plants/progenies to be measured. In this study, we evaluated the dataset distribution and the descriptive statistics of CVe through the Frequentist and Bayesian approaches, aiming to establish the credibility and confidence intervals. We submitted CVe data of ten wheat (Triticum aestivum L.) traits reported in 1,068 articles published to the Bayesian and Frequentist analyses. Sample data were analyzed via Gamma and normal models. We selected the model with the lowest Akaike Information Criterion (AIC) value, and then we tested three link functions. In the Bayesian analysis, uniform distributions were used as non-informative priors for the Gamma distribution parameters with three ranges of q~U (a,b,). Thus, the prior probability density function was given by: pθ =1β-α, θ ∈ α,β. The Bayesian and Frequentist approaches with the Gamma model presented similar results for CVe; however, the range Bayesian credible intervals was narrower than the Frequentist confidence intervals. Gamma distribution fitted the CVe data better than the normal distribution. The credible and confidence intervals of CVe were successfully applied to wheat traits and could be used as experimental accuracy measurements in other experiments.

An improved GNSS/INS navigation method based on cubature Kalman filter for occluded environment

Global navigation satellite systems and inertial navigation systems (GNSSs/INSs) based on the cubature Kalman filter (CKF) can provide an effective land vehicle navigation and positioning solution. The accuracy of CKF depends mainly on the prior probability density function (PDF), which is inaccurate when the GNSS signal is temporarily blocked in an urban environment. Although the improved CKF (ICKF) can reduce cubature point errors by using the historical information obtained iteratively, the generation of cubature points is still related to measurement noise. To obtain a more accurate PDF, adjustment off the sensor measurement noise is necessary. In this paper, an adaptive ICKF for GNSS/INS system based on the Mahalanobis distance called MD-ICKF is proposed, which expands the usability of the micro-electronic mechanical system based integrated navigation system in an urban environment. Two lower triangular matrices obtained by Cholesky decomposition are used to adjust the cubature point errors during each iteration. Furthermore, the measurement noise will be inflated to obtain a more accurate PDF when it exceeds the threshold calculated by the MD and chi-square test. The field experiment results show that the proposed method has better positioning accuracy and performance than CKF and ICKF in an occluded environment

A probabilistic model on streamwise velocity profile in open channels using Tsallis relative entropy theory

The present study uses the concept of information entropy to derive a velocity distribution model in open channel flow. The Tsallis relative entropy theory is explored for that purpose, where the prior probability density function (PDF) is chosen from the maximum Tsallis entropy distribution. Both the cases of the fixed and varying entropy index are considered from the literature. The velocity equations are derived analytically using the homotopy analysis method (HAM) together with the Padé approximation technique in a general framework. The approximations and the HAM-based series solution are verified against laboratory and field data sets. Also, the prediction accuracies of the proposed models are assessed through the measures of statistical errors. It is seen that the model corresponding to the varying index is superior to the other model. The entropy index of Tsallis relative entropy is considered a variable, and the non-unity values of this index justify the applicability of the entropy function. Moreover, it is observed that one of the Lagrange multipliers corresponding to the velocity model with a varying index becomes negligible subject to the data sets considered, and hence, it simplifies the model development mathematically. The proposed approach can be further extended to develop models for estimating the suspended sediment concentration and shear stress distribution.

Improved unscented Kalman filter for state of charge estimation of lithium-ion battery with one-step randomly measurement loss and inaccurate noise covariance matrices

Accurate state of charge estimation is of great significance for optimizing the management of lithium batteries. Due to the influence of actual environmental factors, random measurement loss and inaccurate noise covariance matrices will appear in the estimation process. Especially when the two situations occur at the same time, the estimation of SOC will be seriously affected. Thus, an improved unscented Kalman filter is presented to solve the above problems in this paper. First, the state space model of nonlinear system is constructed, which can fully reflect one-step random measurement loss, and the nominal noise covariance matrix is used to reflect the uncertainty. Second, as the prior probability density function of the parameters, the Beta prior and inverse Wishart prior are used in the derivation of the joint probability density function. Finally, these parameters are derived based on variational Bayesian theory, the updated state vector is obtained. Simulation results shows that the proposed method can deal with random measurement loss and inaccurate noise covariance matrices and the high accuracy in dynamic stress test and federal urban driving schedule.

Plate‐Locking, Uncertainty Estimation and Spatial Correlations Revealed With a Bayesian Model Selection Method: Application to the Central Chile Subduction Zone

AbstractInversions of geodetic data are regularly used to estimate interseismic locking in subduction zones. However, the ill‐posed nature of these problems motivates us to include prior information, physically consistent with processes of the subduction seismic cycle. To deal with model instabilities, we present an inversion method to estimate both plate‐locking and model uncertainties by inverting Global Navigation Satellite System derived velocities based on a Bayesian model selection scheme. Our method allows us to impose positivity constraints via a multivariate folded‐normal distribution, with a specified covariance matrix. Model spatial correlations are explored and ranked to find models that best explain the observed data and for a better understanding of locking models. This approach searches for hyperparameters of the prior joint multivariate probability density function (PDF) of model parameters that minimize the Akaike Bayesian Information Criterion (ABIC). To validate our approach, we invert synthetic displacements from analytic models, yielding satisfactory results. We then apply the method to estimate the plate‐locking in Central Chile (28°–39°S) and its relation to the coseismic slip distribution of earthquakes with magnitudes Mw > 8.0, on the subduction zone since 2010. We also search among different prior PDFs for a single ductile‐fragile limit depth. Our results confirms a spatial correlation between locked asperities and the 2010 Mw 8.8 Maule and 2015 Mw 8.3 Illapel earthquake rupture zones. The robustness of our locking model shows potential to improve future seismic and tsunami hazard estimations.

Explosive source signature determination: Two unequal shots in the same hole

AbstractBuried explosive charges generate seismic waves with unrivalled bandwidth. The source time function is not always repeatable and is difficult to measure, but is required for subsequent data processing including reverse time migration and full waveform inversion. I present a new method for estimating the explosive seismic source time function for every shot, using two unequal shots in the same shot hole. The measured data for each charge are deconvolved for the modelled monopole seismic source time function to recover estimated earth impulse responses or Green's functions. The source model parameters are adjusted to maximize the likeness of the estimated Green's functions from the two shots, applying physical constraints to obtain the prior probability density functions for each parameter. Ten parameters specify the two source models, but only five of these are independent and a grid search is used to find them. The method is tested on an experimental data set, obtained for a different purpose, consisting of a line of single geophones and three in‐line shot holes, each with two unequal charges, the larger one at twice the depth of the smaller. The method works well where the charge size ratio is at least 2: the estimated source time functions are recovered and the estimated Green's functions are more similar than the seismograms before deconvolution. To apply the method in practice, the smaller charge should be deeper than the larger. If the small charge is small enough, the vertical separation between the charges can be reduced to less than 1 m. The Green's functions for the two charges would then be almost identical for frequencies up to 250 Hz, and the differences in the measured data from the two charges would be caused principally by the differences in the two source time functions.

Bayesian Multi‐Model Estimation of Fault Slip Distribution for Slow Slip Events in Southwest Japan: Effects of Prior Constraints and Uncertain Underground Structure

AbstractWe consider a Bayesian multi‐model fault slip estimation (BMMFSE), which incorporates many candidates of the underground structure (Earth structure and plate boundary geometry) model characterized by a prior probability density function (PDF). The technique is used to study long‐term slow slip events (L‐SSEs) that occurred beneath the Bungo Channel, southwest Japan, in around 2010 and 2018. We here focus on the two advantages of BMMFSE: First, it allows for estimating slip distribution without introducing relatively strong prior information such as smoothing constraints, by combining a fully Bayesian inference and better consideration of model uncertainty to avoid overfitting. Second, the posterior PDF for the underground structure is also obtained during the fault slip estimation, which can be used as priors for the estimation of slip distribution for recurring events. The estimated slip distribution obtained using BMMFSE agreed better with the distribution of deep tectonic tremors at the down‐dip side of the main rupture area than those based on stronger prior constraints when the corresponding Coulomb failure stress changes are compared. This finding suggests a mechanical relationship between the L‐SSE and the synchronized tremors. The use of the posterior PDF of the underground structure estimated for the 2010 L‐SSE as prior PDF for the 2018 event resulted in more consistent estimation with the data, indicated by a smaller value of an information criterion.

Machine learning‐accelerated gradient‐based Markov chain Monte Carlo inversion applied to electrical resistivity tomography

ABSTRACTExpensive forward model evaluations and the curse of dimensionality usually hinder applications of Markov chain Monte Carlo algorithms to geophysical inverse problems. Another challenge of these methods is related to the definition of an appropriate proposal distribution that simultaneously should be inexpensive to manipulate and a good approximation of the posterior density. Here we present a gradient‐based Markov chain Monte Carlo inversion algorithm that is applied to cast the electrical resistivity tomography into a probabilistic framework. The sampling is accelerated by exploiting the Hessian and gradient information of the negative log‐posterior to define a proposal that is a local, Gaussian approximation of the target posterior probability. On the one hand, the computing time to run the many forward evaluations needed for both the data likelihood evaluation and the Hessian and gradient computation is decreased by training a residual neural network to predict the forward mapping between the resistivity model and the apparent resistivity value. On the other hand, the curse of dimensionality issue and the computational effort related to the Hessian and gradient manipulation are decreased by compressing data and model spaces through a discrete cosine transform. A non‐parametric distribution is assumed as the prior probability density function. The method is first demonstrated on synthetic data and then applied to field measurements. The outcomes provided by the presented approach are also benchmarked against those obtained when a computationally expensive finite‐element code is employed for forward modelling , with the results of a gradient‐free Markov chain Monte Carlo inversion, and also compared with the predictions of a deterministic inversion. The implemented approach not only guarantees uncertainty assessments and model predictions comparable with those achieved by more standard inversion strategies, but also drastically decreases the computational cost of the probabilistic inversion, making it similar to that of a deterministic inversion.

A Quantile-Conserving Ensemble Filter Framework. Part I: Updating an Observed Variable

Abstract A general framework for deterministic univariate ensemble filtering is presented. The framework fits a continuous prior probability density function (PDF) to the prior ensemble. A functional representation for the observation likelihood is combined with the prior PDF to get a continuous analysis (posterior) PDF. Cumulative distribution functions for the prior and analysis are also required. The key innovation is that an analysis ensemble is computed so that the quantile of each ensemble member is the same as its prior quantile. Many choices for the prior PDF family and the likelihood function are described. A choice of normal prior with normal likelihood is equivalent to the ensemble adjustment Kalman filter. Some other choices for the prior include gamma, inverse gamma, beta, beta prime, lognormal, and exponential distributions. Both prior distributions and likelihoods can be defined over a set of intervals giving additional flexibility that can be used to implement methods like a Huber likelihood for observations with occasional outliers. Priors and likelihoods can also be defined as sums of distributions allowing choices like bivariate normals or kernel filters. Empirical distributions, for instance piecewise linear approximations to arbitrary PDFs and functions can be used. Another empirical choice leads to the rank histogram filter. Results here are univariate and can be used to compute increments for observed variables or marginal distributions for any variable for a reanalysis. Linear regression of increments can be used to update state variables in a serial filter to build a comprehensive data assimilation system. Part 2 will discuss other methods for extending the framework to multivariate data assimilation. Significance Statement Data assimilation is used to combine information from model forecasts with subsequent observations to obtain better estimates of the current state of the atmosphere or other parts of the Earth system. Ensemble data assimilation uses a number of forecasts to get more information about uncertainty. A new method allows much more flexibility in the assumptions that must be made when doing ensemble data assimilation. As an example, the method can be better for quantities that are bounded like the amount of an atmospheric trace pollutant.

Deep Residual Learning for Channel Estimation in Intelligent Reflecting Surface-Assisted Multi-User Communications

Channel estimation is one of the main tasks in realizing practical intelligent reflecting surface-assisted multi-user communication (IRS-MC) systems. However, different from traditional communication systems, an IRS-MC system generally involves a cascaded channel with a sophisticated statistical distribution. In this case, the optimal minimum mean square error (MMSE) estimator requires the calculation of a multidimensional integration which is intractable to be implemented in practice. To further improve the channel estimation performance, in this paper, we model the channel estimation as a denoising problem and adopt a deep residual learning (DReL) approach to implicitly learn the residual noise for recovering the channel coefficients from the noisy pilot-based observations. To this end, we first develop a versatile DReL-based channel estimation framework where a deep residual network (DRN)-based MMSE estimator is derived in terms of Bayesian philosophy. As a realization of the developed DReL framework, a convolutional neural network (CNN)-based DRN (CDRN) is then proposed for channel estimation in IRS-MC systems, in which a CNN denoising block equipped with an element-wise subtraction structure is specifically designed to exploit both the spatial features of the noisy channel matrices and the additive nature of the noise simultaneously. In particular, an explicit expression of the proposed CDRN is derived and analyzed in terms of Bayesian estimation to characterize its properties theoretically. Finally, simulation results demonstrate that the performance of the proposed method approaches that of the optimal MMSE estimator requiring the availability of the prior probability density function of channel.

A hybrid framework for remaining useful life estimation of turbomachine rotor blades

Hybrid methods for prognosis of mechanical components have the potential of improving remaining useful life (RUL) estimations. Hybrid methods combine physics-based and data-driven methods to diagnose faults and predict when failures will occur. In this work, we propose a hybrid framework that estimates the RUL from routine maintenance inspection data and condition monitoring data.This hybrid framework is applied to turbomachine rotor blades. Blade tip timing (BTT) measurements are used for condition monitoring. The least squares spectral analysis (LSSA) method is used to find the natural frequency. The natural frequency is a function of the blade's health state and is used to infer the crack length in the blade’s root. To accommodate for artificial rotational stiffening, an interpolation model of the blade's Campbell diagram, generated from a finite element model, is used.In the proposed methodology, we use an ensemble physics-based model that serves as a prior probability density function for a Gaussian process regression (GPR) model. The predictive distribution of the GPR model is constructed by conditioning the physics-based model on the observed data from routine maintenance inspections. We also compare this hybrid diagnosis model to the physics-based technique and other data-driven methods. The hybrid diagnosis model outperformed the physics-based method and data-driven methods.The unscented Kalman filter (UKF) is used to estimate and forecast the evolution of the crack length over time. Paris’ law coefficients are used as hidden latent variables in a hybrid degradation model. In the diagnosis and prognosis phases, we use the unscented transform to efficiently approximate the probability density functions of the crack length and the crack growth law parameters.Finally, we compare the applied hybrid model to a purely physics-based model and show that the hybrid model outperforms the physics-based model in predicting the length of a crack. We demonstrate that the hybrid model increases the accuracy and precision of RUL prediction from physics-based models with 60% on average.

An improved unscented Kalman filter for nonlinear systems with one-step randomly delayed measurement and unknown latency probability

In this paper, an improved unscented Kalman filter is presented to achieve state estimation for nonlinear systems with one-step randomly delayed measurement and unknown latency probability. Firstly, a Bernoulli random variable is introduced to describe the situation of randomly delayed measurement. Then, a joint prior probability density function is obtained. Finally, in the improved unscented Kalman filter algorithm, the state vector and the latency probability are estimated by utilizing a variational Bayesian technique, and the covariance matrices are computed by using the unscented transform. A univariate nonstationary growth model and a Lorenz system are included to verify that the proposed algorithm can not only implement enhanced estimation accuracy compared with the existing methods but also provide the latency probability accurately.

Quantification of the uncertainty within a SAS-SST simulation caused by the unknown high-wavenumber damping factor

This paper aims to quantify the uncertainty in the SAS-SST simulation of a prism bluff-body flow due to varying the higher-wavenumber damping factor (Cs). Instead of performing the uncertainty quantification on the CFD simulation directly, a surrogate modelling approach is adopted. The mesh sensitivity is first studied and the numerical error due to the mesh is approximated accordingly. The Gaussian processes/Kriging method is used to generate surrogate models for quantities of interest (QoIs). The suitability of the surrogate models is assessed using the leave-one-out cross-validation tests (LOO-CV). The stochastic tests are then performed using the cross-validated surrogate models to quantify the uncertainty of QoIs by varying Cs. Four prior probability density functions (such as U0,1, N0.5,0.12, Beta2,2 and Beta5,1.5) of Cs are considered.It is demonstrated in this study that the uncertainty of a predicted QoI due to varying Cs is regionally dependent. The flow statistics in the near wake of the prism body are subject to larger variance due to the uncertainty in Cs. The influence of Cs rapidly decays as the location moves downstream. The response of different QoIs to the changing Cs varies greatly. Therefore, the calibration of Cs only using observations of one variable may bias the results. Last but not least, it is important to consider different sources of uncertainties within the numerical model when scrutinising a turbulence model, as ignoring the contributions to the total error may lead to biased conclusions.

A unified sampling-based framework for optimal sensor placement considering parameter and prediction inference

We present a Bayesian framework for model-based optimal sensor placement. Our interest lies in minimizing the uncertainty on predictions of a particular response quantity of interest, with parameter estimation being an intermediate step for this purpose. By developing a methodology that targets prediction inference rather than parameter inference, we prioritize reduction of uncertainty on the parameters that matter most for the prediction of the actual quantity of interest. Currently available optimal sensor placement methods focus on parameter inference rather than prediction inference and might therefore yield suboptimal solutions for prediction inference. We opt for a unifying framework where the case of parameter inference is merely a special case of prediction inference. Following the Bayesian framework for uncertainty quantification, the model parameters are treated as random variables and their uncertainty before data collection is described by a prior probability density function. The prior uncertainty is updated to the posterior uncertainty using measured data that depends on the chosen sensor locations. This posterior parameter uncertainty is then converted to the posterior prediction uncertainty. As a scalar measure of uncertainty, we use the determinant of the posterior prediction covariance matrix. This is a general type of metric which can be used for both prediction and parameter inference. Using the expectation of this determinant with respect to the distribution of possible data as the objective function, the sensor locations are optimized to minimize the expected parameter or prediction uncertainty. The required covariance matrices of parameters and predictions are evaluated using a Monte Carlo sampling approach. We verify this procedure for a simple test example and present a (simplified) case study from structural dynamics where sensor locations in a modal test are optimized for parameter and prediction inference. We show how the optimal locations for prediction uncertainty differ from those obtained by minimizing parameter uncertainty. In general, the difference will depend on the prior parameter uncertainties, the way the experimental data depend on the parameters, and the way the predictions depend on the parameters. Significant differences will occur when the data as well as the predictions are local in nature and optimizing for prediction inference allows adapting the data such that they are most informative for the relevant parameter subset.

Algorithms and Analysis for Optimizing the Tracking Performance of Cyber Attacked Sensor-Equipped Connected Vehicle Networks

Sensor-equipped connected vehicle networks (SECVNs) have the potential to enable substantially safer driving by improved object tracking, which is an important basic building block in SECVNs. Unfortunately, cyber-attacks on SECVNs pose a very serious threat which could lead to unacceptable outcomes, including fatalities. Recently there has been increasing focus on malicious attack detection and mitigation in SECVNs, and some of this work has considered attacks on sensor data to impact object tracking. Unfortunately, low complexity mitigation approaches which do not compromise performance are lacking. This paper describes an efficient machine-learning enhanced approach for tracking under cyber-attacks. By proper selection of some variances related to the sensor and prior probability density functions, under some assumptions the performance can be made as close as desired to a bound on the best possible performance. However, the complexity of this new approach is dramatically lower than the best existing published low complexity approach, which provides performance which is substantially inferior to that provided by the new approach. The new approach also provides much better scaling with the size of the SECVN. In particular, the complexity increases linearly in the number of sensors, while the best low complexity published approach has a complexity which grows quadratically in the number of sensors. The new approach is also applicable to other tracking applications.